2,962 research outputs found
A new algorithm for recognizing the unknot
The topological underpinnings are presented for a new algorithm which answers
the question: `Is a given knot the unknot?' The algorithm uses the braid
foliation technology of Bennequin and of Birman and Menasco. The approach is to
consider the knot as a closed braid, and to use the fact that a knot is
unknotted if and only if it is the boundary of a disc with a combinatorial
foliation. The main problems which are solved in this paper are: how to
systematically enumerate combinatorial braid foliations of a disc; how to
verify whether a combinatorial foliation can be realized by an embedded disc;
how to find a word in the the braid group whose conjugacy class represents the
boundary of the embedded disc; how to check whether the given knot is isotopic
to one of the enumerated examples; and finally, how to know when we can stop
checking and be sure that our example is not the unknot.Comment: 46 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTVol2/paper9.abs.htm
Alexander representation of tangles
A tangle is an oriented 1-submanifold of the cylinder whose endpoints lie on
the two disks in the boundary of the cylinder. Using an algebraic tool
developed by Lescop, we extend the Burau representation of braids to a functor
from the category of oriented tangles to the category of Z[t,t^{-1}]-modules.
For (1,1)-tangles (i.e., tangles with one endpoint on each disk) this invariant
coincides with the Alexander polynomial of the link obtained by taking the
closure of the tangle. We use the notion of plat position of a tangle to give a
constructive proof of invariance in this case.Comment: 13 pages, 5 figure
Smilansky's model of irreversible quantum graphs, II: the point spectrum
In the model suggested by Smilansky one studies an operator describing the
interaction between a quantum graph and a system of K one-dimensional
oscillators attached at different points of the graph. This paper is a
continuation of our investigation of the case K>1. For the sake of simplicity
we consider K=2, but our argument applies to the general situation. In this
second paper we apply the variational approach to the study of the point
spectrum.Comment: 18 page
Penn State Get Away Special
Three proposed spaceborne experiments to be conducted by equipment in the Get Away Special (GAS) payload are described. The specific contribution and effect of convection in heat transfer is discussed. Investigations of the surface tension of two liquids in space environment and the problem of liquid slosh in spin stabilized satellites are reviewed
Quasi-classical versus non-classical spectral asymptotics for magnetic Schroedinger operators with decreasing electric potentials
We consider the Schroedinger operator H on L^2(R^2) or L^2(R^3) with constant
magnetic field and electric potential V which typically decays at infinity
exponentially fast or has a compact support. We investigate the asymptotic
behaviour of the discrete spectrum of H near the boundary points of its
essential spectrum. If the decay of V is Gaussian or faster, this behaviour is
non-classical in the sense that it is not described by the quasi-classical
formulas known for the case where V admits a power-like decay.Comment: Corrected versio
Spectral and scattering theory of fourth order differential operators
An ordinary differential operator of the fourth order with coefficients
converging at infinity sufficiently rapidly to constant limits is considered.
Scattering theory for this operator is developed in terms of special solutions
of the corresponding differential equation. In contrast to equations of second
order "scattering" solutions contain exponentially decaying terms. A relation
between the scattering matrix and a matrix of coefficients at exponentially
decaying modes is found. In the second part of the paper the operator on
the half-axis with different boundary conditions at the point zero is studied.
Explicit formulas for basic objects of the scattering theory are found. In
particular, a classification of different types of zero-energy resonances is
given
Spin networks, quantum automata and link invariants
The spin network simulator model represents a bridge between (generalized)
circuit schemes for standard quantum computation and approaches based on
notions from Topological Quantum Field Theories (TQFT). More precisely, when
working with purely discrete unitary gates, the simulator is naturally modelled
as families of quantum automata which in turn represent discrete versions of
topological quantum computation models. Such a quantum combinatorial scheme,
which essentially encodes SU(2) Racah--Wigner algebra and its braided
counterpart, is particularly suitable to address problems in topology and group
theory and we discuss here a finite states--quantum automaton able to accept
the language of braid group in view of applications to the problem of
estimating link polynomials in Chern--Simons field theory.Comment: LateX,19 pages; to appear in the Proc. of "Constrained Dynamics and
Quantum Gravity (QG05), Cala Gonone (Italy) September 12-16 200
Cosmic Strings Stabilized by Fermion Fluctuations
We provide a thorough exposition of recent results on the quantum
stabilization of cosmic strings. Stabilization occurs through the coupling to a
heavy fermion doublet in a reduced version of the standard model. The study
combines the vacuum polarization energy of fermion zero-point fluctuations and
the binding energy of occupied energy levels, which are of the same order in a
semi-classical expansion. Populating these bound states assigns a charge to the
string. Strings carrying fermion charge become stable if the Higgs and gauge
fields are coupled to a fermion that is less than twice as heavy as the top
quark. The vacuum remains stable in the model, because neutral strings are not
energetically favored. These findings suggest that extraordinarily large
fermion masses or unrealistic couplings are not required to bind a cosmic
string in the standard model.Comment: Based on talk by HW at QFEXT 11 (Benasque, Spain), 15p, uses
ws-ijmpcs.cls (incl
Topologically protected quantum gates for computation with non-Abelian anyons in the Pfaffian quantum Hall state
We extend the topological quantum computation scheme using the Pfaffian
quantum Hall state, which has been recently proposed by Das Sarma et al., in a
way that might potentially allow for the topologically protected construction
of a universal set of quantum gates. We construct, for the first time, a
topologically protected Controlled-NOT gate which is entirely based on
quasihole braidings of Pfaffian qubits. All single-qubit gates, except for the
pi/8 gate, are also explicitly implemented by quasihole braidings. Instead of
the pi/8 gate we try to construct a topologically protected Toffoli gate, in
terms of the Controlled-phase gate and CNOT or by a braid-group based
Controlled-Controlled-Z precursor. We also give a topologically protected
realization of the Bravyi-Kitaev two-qubit gate g_3.Comment: 6 pages, 7 figures, RevTeX; version 3: introduced section names, new
reference added; new comment added about the embedding of the one- and two-
qubit gates into a three-qubit syste
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